To determine the rule used to translate the triangle, follow these steps:
1. Identify the original coordinates of point [tex]\( L \)[/tex] and its translated coordinates [tex]\( L' \)[/tex]:
- [tex]\( L(7,-3) \)[/tex]
- [tex]\( L'(-1,8) \)[/tex]
2. Calculate the translation vector. This vector represents how much to move in the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-directions to get from [tex]\( L \)[/tex] to [tex]\( L' \)[/tex].
- The translation in the [tex]\( x \)[/tex]-direction is given by the difference in the [tex]\( x \)[/tex]-coordinates: [tex]\( -1 - 7 = -8 \)[/tex].
- The translation in the [tex]\( y \)[/tex]-direction is given by the difference in the [tex]\( y \)[/tex]-coordinates: [tex]\( 8 - (-3) = 8 + 3 = 11 \)[/tex].
3. Thus, the translation vector is [tex]\( (-8, 11) \)[/tex].
4. Translate this vector into the form of a translation rule. The rule expresses how to move any point [tex]\( (x, y) \)[/tex] to its new position:
- The translation rule derived from the vector [tex]\( (-8, 11) \)[/tex] is [tex]\( (x, y) \rightarrow (x-8, y+11) \)[/tex].
5. Compare the derived translation rule with the given choices:
- [tex]\( (x, y) \rightarrow (x+6, y-5) \)[/tex]
- [tex]\( (x, y) \rightarrow (x-6, y+5) \)[/tex]
- [tex]\( (x, y) \rightarrow (x+8, y-11) \)[/tex]
- [tex]\( (x, y) \rightarrow (x-8, y+11) \)[/tex]
6. The correct match is:
- [tex]\( (x, y) \rightarrow (x-8, y+11) \)[/tex]
Therefore, the rule used to translate triangle [tex]\( LMN \)[/tex] so that [tex]\( L \)[/tex] translates to [tex]\( L' \)[/tex] is:
[tex]\[
(x, y) \rightarrow (x-8, y+11)
\][/tex]
The correct answer is:
[tex]\[
4
\][/tex]
